It would be counter-productive for sure. The term 'Googling' has already been adopted by vocabularies worldwide as a verb. It's separating itself more and more from the actual brand. Something which companies hate! Like Luxaflex, Elevator, Bubble Wrap, Jet Ski - it's a brand marketeer's dream and worst nightmare all at once:
http://mentalfloss.com/article/56667/41-brand-names-people-use-generic-terms
If Google missteps somewhere in the future we will 'Google' on DuckDuckGo, Bing or who-ever offers the same quality without the annoyances![]()